Sets and subsets, Need help ASAP please!

[confuzzled]

There are 4 questions I just cant get in my homework and I will need to be able to do these on a test in 2 days, any help anyone could give me would be awesome. Thanks!

The first 3 I think are giving me trouble because the variables are present in place of actual numbers and it confuses me. I've thought about them for hours and I dont understand at all how to start them.

1. A subset of size r is formed from a set of size n that includes the symbols A and B. How many of these subsets contain A or B?

2. Given a set of n symbols, how many sequences of length 2r (where 2r<n) can be formed such that every symbol used is included in the sequence twice? (Eg. A sequence of length 8 could be RABACCBR or RRAABBCC, etc..)

3. How many binary sequences of length 2n have the same number of 1s in the first and second half of the sequence?


4. Determine n(AUB) such that:
A= set of sequences of length 3 made from the letters GEOMETRY
B= set of sequences of length 3 made from the letters DISCRETE

number 4 messes me up because I dont know what to use to get the n(A) & n(B). I dont know whether or not to count the duplication of the E's..or to only count them once.. is it just 8! or do I have to use combinations or something?

 

[pka]

1. A subset of size r is formed from a set of size n that includes the symbols A and B. How many of these subsets contain A or B?

2. Given a set of n symbols, how many sequences of length 2r (where 2r<n) can be formed such that every symbol used is included in the sequence twice? (Eg. A sequence of length 8 could be RABACCBR or RRAABBCC, etc..)

3. How many binary sequences of length 2n have the same number of 1s in the first and second half of the sequence?

4. Determine n(AUB) such that:
A= set of sequences of length 3 made from the letters GEOMETRY
B= set of sequences of length 3 made from the letters DISCRETE

These questions range for trivial to very difficult.

#1 is a simple inclusion/exclusion: #(A)+#(B)-#(A and B).

I really do not understand #2. That is, what are the strings?

#3 is the most interesting of the four. Suppose \(\displaystyle 0 \le r \le n\), then if we count the number of n-bit strings that have exactly r 1's. Each one of those strings can be paired with itself or with any other of those to have one of the strings #3 is asking for.
Thus the answer is: \(\displaystyle \L \sum\limits_{k = 0}^n {\left( {\begin{array}{c}
n \\
k \\
\end{array}} \right)^2 }.\)

For #4, YES you must take into consideration the double E’s.
#(A)=#(B) so consider GEOMETRY. If we remove one E, GOMETRY, there are seven letters to arrange into 3-strings. That can be done in (7)(6)(5) ways. That is also the number of such strings that contain at most one E. There are 3(6) 3-strings that contain two E’s: use two E’s and choose any one of the six others. Then for example GEE can be arranged in three ways. #(A)=#(B)= (7)(6)(5)+3(6). However, the two words share some letters, <EERT>. So we have counted some 3-strings twice.
{E,R,T} accounts for 6 of those; <EER> accounts for 3; so does <EET>. That give us twelve 3-strings they have in common.